Generalization bounds for nonparametric regression with $\beta-$mixing samples

• URL: http://arxiv.org/abs/2108.00997v1
• Date: Mon, 2 Aug 2021 15:51:52 GMT
• Title: Generalization bounds for nonparametric regression with $\beta-$mixing samples
• Authors: David Barrera and Emmanuel Gobet
• Abstract summary: We present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case. We then apply these results to some previously obtained inequalities for independent samples associated to the deviation of the least-squared error in nonparametric regression.
• Score: 3.680403821470857
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper we present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case characterizing the additional error in terms of $\beta-$mixing coefficients associated to the training sample. We then apply these results to some previously obtained inequalities for independent samples associated to the deviation of the least-squared error in nonparametric regression to derive corresponding generalization bounds for regression schemes in which the training sample may not be independent. These results provide a framework to analyze the error associated to regression schemes whose training sample comes from a large class of $\beta-$mixing sequences, including geometrically ergodic Markov samples, using only the independent case. More generally, they permit a meaningful extension of the Vapnik-Chervonenkis and similar theories for independent training samples to this class of $\beta-$mixing samples.

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